Finite Group Databasebeta

← #192472

Subgroups — #192472

Order 512.

1033 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2976C211yesmaximal, characteristic
2#21949C211yesmaximal, characteristic, complemented
3#21932C211yesmaximal, characteristic, complemented
4#21774C211yesmaximal, characteristic
5#17542C211yesmaximal, characteristic
6#8875C214yesmaximal, complemented
7#8874C214yesmaximal, complemented
8#8051C211yesmaximal, characteristic, complemented
9#7927C211yesmaximal, characteristic, complemented
10C2×#525C2211yescharacteristic
11#2976—22no—
12#2976—22no—
13#2976—22no—
14#2976C2214yes—
15C2×#436—21no—
16C2×#435—21no—
17C2×#435—21no—
18C2×C8×D4—21no—
19C2×#433—21no—
20C2×#408—21no—
21C2×#407C2211yescharacteristic
22C2×#407C2211yescharacteristic
23C2×#407C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25#2280C2211yescharacteristic
26C2×C4×C22⋊C4C2211yescharacteristic
27#1897—22no—
28#1871C2212yes—
29#1871C2212yes—
30#1855—21no—
31#1855—21no—
32#1855C2212yescomplemented
33#1855C2212yescomplemented
34#1851C2212yescomplemented
35#1851C2212yescomplemented
36#1850—21no—
37#1850—21no—
38#1826—21no—
39#1811—22no—
40#1797—22no—
41#1789—21no—
42#1508C411yescharacteristic, complemented
43#1508C411yescharacteristic, complemented
44#1507C411yescharacteristic, complemented
45#1507C411yescharacteristic, complemented
46#1504C414yescomplemented
47#1501C2214yes—
48#1297C2214yescomplemented
49#1282C2214yescomplemented
50C22×#41D411yescharacteristic
51C24×C4C2311yescharacteristic
52C22×M16—21no—
53C22×M16—21no—
54C22×M16D411yescharacteristic
55C22×M16D411yescharacteristic
56C22×M16C2311yescharacteristic
57C22×M16C2311yescharacteristic
58C23×C8—21no—
59C23×C8D411yescharacteristic
60C23×C8D411yescharacteristic
61#525—21no—
62#525—21no—
63#525—21no—
64#525—21no—
65#525C2×C412yes—
66#525D412yes—
67#525Q812yes—
68#525C2312yes—
69C2×#123—21no—
70C2×C22⋊Q8C2×C411yescharacteristic
71C2×C22⋊Q8Q811yescharacteristic
72C2×#121C2×C411yescharacteristic
73C2×#121Q811yescharacteristic
74C2×#120—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×#117—21no—
78C2×#117D411yescharacteristic
79C22×C22⋊C4—21no—
80C22×C42—21no—
81#436—24no—
82#436—24no—
83#435—24no—
84#435—24no—
85#435—24no—
86#435—24no—
87C8×D4—24no—
88C8×D4—24no—
89#433—24no—
90#433—24no—
91C2×#108—22no—
92C2×#108—22no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×#104—22no—
98C2×#104—22no—
99#408—24no—
100#408—24no—
101#407—21no—
102#407—21no—
103#407—21no—
104#407—21no—
105#407—21no—
106#407—21no—
107#407—21no—
108#407—21no—
109#407C2×C412yes—
110#407C2×C411yescharacteristic
111#407C2×C411yescharacteristic
112#407D412yes—
113#407C2312yes—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8D412yes—
126C2×C22⋊C8C2312yes—
127C2×C22⋊C8C2312yes—
128C2×#97—21no—
129C2×C4×C8—21no—
130#381—21no—
131#380—21no—
132#380—21no—
133#379—21no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4C2×C411yescharacteristic
138C4×C22⋊C4C2×C411yescharacteristic
139C4×C22⋊C4C2312yes—
140C4×C22⋊C4C2312yes—
141C2×#95C2×C412yes—
142#341—22no—
143#341—22no—
144#339—22no—
145#339—22no—
146#336—22no—
147#336—22no—
148#328C2×C414yes—
149#328C2×C414yes—
150#327C2×C414yes—
151#327C2×C414yes—
152C25#4111yescharacteristic
153C2×#41—21no—
154C2×#41—21no—
155C2×#41—81no—
156C2×#41—81no—
157C2×#41C4⋊C412yes—
158C2×#41C2×D412yes—
159C22×Q8C4⋊C411yescharacteristic
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4C4⋊C411yescharacteristic
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—81no—
167C23×C4—81no—
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×Q811yescharacteristic
177C23×C4C2×Q811yescharacteristic
178C23×C4#4111yescharacteristic
179C23×C4#4111yescharacteristic
180C23×C4#4111yescharacteristic
181C23×C4C2411yescharacteristic
182C2×M16—24no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—22no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—81no—
193C2×M16—81no—
194C2×M16—81no—
195C2×M16—81no—
196C2×M16—81no—
197C2×M16—81no—
198C2×M16—82no—
199C2×M16C2×D412yes—
200C2×M16C2×D412yes—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—22no—
206C22×C8—24no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—81no—
212C22×C8—81no—
213C22×C8—82no—
214C22×C8C2×D412yes—
215C22×C8C2×D412yes—
216#123—22no—
217#123—22no—
218#123—22no—
219#123—22no—
220C22⋊Q8—21no—
221C22⋊Q8—21no—
222C22⋊Q8—21no—
223C22⋊Q8—21no—
224C22⋊Q8C4⋊C412yes—
225C22⋊Q8C4⋊C412yes—
226C22⋊Q8C22×C412yes—
227C22⋊Q8C2×Q812yes—
228#121—21no—
229#121—21no—
230#121—21no—
231#121—21no—
232#121C4⋊C412yes—
233#121C4⋊C412yes—
234#121C22×C412yes—
235#121C2×Q812yes—
236#120—22no—
237#120—22no—
238#120—22no—
239#120—22no—
240C4×D4—22no—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—22no—
247C4×D4—22no—
248#117—21no—
249#117—21no—
250#117—81no—
251#117—81no—
252#117C4⋊C412yes—
253#117C2×D412yes—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4C4⋊C411yescharacteristic
258C2×C4⋊C4C4⋊C411yescharacteristic
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—41no—
271C2×C42—21no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—81no—
277C2×C42—81no—
278#108—82no—
279#108—82no—
280C4⋊C8—24no—
281C4⋊C8—24no—
282C4⋊C8—82no—
283C4⋊C8—82no—
284#104—82no—
285#104—82no—
286C22⋊C8—24no—
287C22⋊C8—24no—
288C22⋊C8—21no—
289C22⋊C8—21no—
290C22⋊C8—22no—
291C22⋊C8—22no—
292C22⋊C8—22no—
293C22⋊C8—22no—
294C22⋊C8—22no—
295C22⋊C8—22no—
296C22⋊C8—22no—
297C22⋊C8—22no—
298C22⋊C8—22no—
299C22⋊C8—22no—
300C22⋊C8—41no—
301C22⋊C8—41no—
302C22⋊C8—41no—
303C22⋊C8—41no—
304C22⋊C8C22×C412yes—
305C22⋊C8C22×C412yes—
306#97—24no—
307C4×C8—24no—
308#95—22no—
309#95—22no—
310#95—41no—
311#95C22×C412yes—
312#95C22×C412yes—
313C24—21no—
314C24—42no—
315C24—42no—
316C24—81no—
317C24C4×D411yescharacteristic
318C24C4×D411yescharacteristic
319C24C22⋊Q811yescharacteristic
320C24C22⋊Q811yescharacteristic
321C24#12311yescharacteristic
322C24#12311yescharacteristic
323C24C2×#4111yescharacteristic
324#41—42no—
325#41—42no—
326#41—81no—
327#41—81no—
328#41—82no—
329#41—82no—
330C2×Q8—21no—
331C2×Q8—81no—
332C2×Q8C2×C4⋊C412yes—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—21no—
340C2×D4—44no—
341C2×D4—81no—
342C2×D4—82no—
343C2×D4—82no—
344C2×D4C2×C4⋊C412yes—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—42no—
371C22×C4—81no—
372C22×C4—81no—
373C22×C4—81no—
374C22×C4—81no—
375C22×C4—81no—
376C22×C4—81no—
377C22×C4—81no—
378C22×C4—82no—
379C22×C4C2×C4⋊C411yescharacteristic
380C22×C4C2×C4⋊C411yescharacteristic
381C22×C4C2×C4⋊C411yescharacteristic
382C22×C4C2×C4⋊C411yescharacteristic
383C22×C4C4×D411yescharacteristic
384C22×C4C4×D411yescharacteristic
385C22×C4C4×D411yescharacteristic
386C22×C4C4×D411yescharacteristic
387C22×C4C4×D411yescharacteristic
388C22×C4C4×D411yescharacteristic
389C22×C4#12011yescharacteristic
390C22×C4#12011yescharacteristic
391C22×C4#12011yescharacteristic
392C22×C4#12011yescharacteristic
393C22×C4C22⋊Q811yescharacteristic
394C22×C4C22⋊Q811yescharacteristic
395C22×C4C22⋊Q811yescharacteristic
396C22×C4C22⋊Q811yescharacteristic
397C22×C4C22⋊Q811yescharacteristic
398C22×C4C22⋊Q811yescharacteristic
399C22×C4#12311yescharacteristic
400C22×C4#12311yescharacteristic
401C22×C4C23×C411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×Q811yescharacteristic
406C22×C4C2×#4111yescharacteristic
407M16—42no—
408M16—42no—
409M16—42no—
410M16—42no—
411M16—42no—
412M16—42no—
413M16—81no—
414M16—81no—
415M16—82no—
416M16—81no—
417M16—81no—
418M16—84no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—81no—
438C2×C8—81no—
439C2×C8—84no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—82no—
447C4⋊C4C2×C4⋊C412yes—
448C4⋊C4C2×C4⋊C412yes—
449C22⋊C4—24no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—44no—
461C22⋊C4—42no—
462C22⋊C4—44no—
463C22⋊C4—82no—
464C22⋊C4—82no—
465C42—22no—
466C42—22no—
467C42—42no—
468C42—42no—
469C42—42no—
470C42—42no—
471C42—81no—
472C42—81no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—88no—
499C23#38511yescharacteristic
500C23#38511yescharacteristic
501C23#38511yescharacteristic
502C23#38511yescharacteristic
503C23C2×C4×D411yescharacteristic
504C23C2×C22⋊Q811yescharacteristic
505C23C2×#12311yescharacteristic
506Q8—42no—
507Q8—81no—
508D4—42no—
509D4—44no—
510D4—44no—
511D4—81no—
512D4—82no—
513D4—82no—
514D4—82no—
515D4—82no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—88no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4#38511yescharacteristic
571C2×C4#38511yescharacteristic
572C2×C4#38511yescharacteristic
573C2×C4#38511yescharacteristic
574C2×C4C22×C4⋊C411yescharacteristic
575C2×C4C2×C4×D411yescharacteristic
576C2×C4C2×#12011yescharacteristic
577C2×C4C2×C22⋊Q811yescharacteristic
578C8—42no—
579C8—42no—
580C8—42no—
581C8—42no—
582C8—42no—
583C8—42no—
584C8—81no—
585C8—81no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—44no—
606C22—44no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—84no—
614C22—82no—
615C22—82no—
616C22#186112yes—
617C22C2×#38511yescharacteristic
618C22#295512yes—
619C4—21no—
620C4—21no—
621C4—21no—
622C4—21no—
623C4—21no—
624C4—21no—
625C4—21no—
626C4—44no—
627C4—42no—
628C4—42no—
629C4—42no—
630C4—82no—
631C4—162no—
632C4#302611yescharacteristic
633C4#302611yescharacteristic
634C2—22no—
635C2—22no—
636C2—41no—
637C2—41no—
638C2—41no—
639C2—41no—
640C2—41no—
641C2—82no—
642C2C2×#186111yescharacteristic
643C2#2088411yescharacteristic
644C2#2088911yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#385
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21949C2semidirect
3. #21932C2semidirect
6. #8875C2semidirect
7. #8874C2semidirect
8. #8051C2semidirect
9. #7927C2semidirect
32. #1855C22semidirect
33. #1855C22semidirect
34. #1851C22semidirect
35. #1851C22semidirect
42. #1508C4semidirect
43. #1508C4semidirect
44. #1507C4semidirect
45. #1507C4semidirect
46. #1504C4semidirect
48. #1297C22semidirect
49. #1282C22semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#29762
#219492
#219322
#217742
#175422
#88752
#88742
#80512
#79272